Density of states in an optical speckle potential

We study the single-particle density of states of a one-dimensional speckle potential, which is correlated and non-Gaussian. We consider both the repulsive and the attractive cases. The system is controlled by a single dimensionless parameter determined by the mass of the particle, the correlation length, and the average intensity of the field. Depending on the value of this parameter, the system exhibits different regimes, characterized by the localization properties of the eigenfunctions. We calculate the corresponding density of states using the statistical properties of the speckle potential. We find good agreement with the results of numerical simulations.

Figure 1

The typical 1D (repulsive) blue-detuned speckle profile. The intensity is given in units of $I_0$ and the coordinate in units of $\xi$.

Figure 2

Density $\rho (I/I_0)$ of minima (top) and maxima (bottom) for a blue-detuned speckle (in log scale in the insets). The solid lines represent the analytical prediction of Eq. (12), while the dots have been obtained by averaging over $100$ numerical speckle realizations.

Figure 3

The normalized distributions of distance measured in units of the autocorrelation length $\xi$ between neighboring maxima (dotted line) and between neighboring minima (solid line).

Figure 4

Probability density distribution of the lowest-lying eigenstates of a red-detuned speckle pattern (bottom panel) for three different intensities corresponding to $s=-0.1,-1,-100$. Only a window of length $150\xi$ of the whole system is shown. The figure shows the ground state (blue solid line) and several first excited states (black dashed line). Note that, as the speckle intensity changes, the position of the eigenstates may change and some may move in or out from the selected window.

Figure 5

Large negative-energy DOS expressed in units of $(\mathrm{\xi ̃}{I}_0){}^{-1}$ in the presence of a red-detuned speckle potential with $s=-2000$ as a function of the rescaled energy $E/I_0$ in linear scale and in logarithmic scale (inset). The dots represent the numerical data (averaged over $100$ realizations), while the solid line indicates the analytical semiclassical curve (18).

Figure 6

The dots represent the numerical data in linear scale and in logarithmic scale (inset) for the low-energy DOS tails in units of $(B{I}_0){}^{-1}$ in a red-detuned speckle potential as a function of $E/I_0$ for different values $s<0$: $s=-2000$ (red squares), $s=-1000$ (blue circles), $s=-1$ (open circles) and $s=-0.1$ (green triangles). The black solid line refers to the analytical semiclassical result (18) valid at large $\left|s\right|$.

Figure 7

The dots represent the DOS for a red-detuned speckle potential with $s=-1$ in logarithmic scale and in linear scale (inset). The solid line describes the modified semiclassical approximation of Eqs. (22, 23, 24) with $\beta =0.5$ valid in the low-energy tail. At positive energies, the DOS approaches asymptotically the 1D continuum ${\nu }_0(E)=\sqrt{m}/\pi \hslash \sqrt{2E}$, indicated by the dashed line.

Figure 8

The dots show the low-energy DOS tail computed numerically for a particle in red-detuned speckle potential with $s=-0.1$. The black solid line corresponds to the variational result of Eq. (27) with $P_{\lambda }[I]$ given by Eq. (25). The inset shows the number of wells occupied by the states as a function of energy. The green squares and red circles are computed for the first few states in a particular speckle sample using the attenuation factor of 10 and 100, respectively. The solid line is the prediction of the variational calculations.

Figure 9

Probability density distribution of the lowest-lying eigenstates of a blue-detuned speckle pattern (bottom panel), for three different intensities with $s=0.1$, 1, 100. Only a window of length $150\xi$ of the whole system is shown. The figure shows the ground state (blue solid line) and several first excited states (black dashed line).

Figure 10

The one-particle DOS for a blue-detuned speckle potential computed numerically for $s=0.1$ (green circles) and $s=1$ (magenta squares). The black solid line is the DOS of free particle. (Inset) The best simultaneous fit of the Lifshitz tail of both DOS by Eq. (36) with the same $c_0=0.8141$ and $A=0.4098$.

Figure 11

The DOS of a particle in a blue-detuned speckle potential are computed numerically for $s=2000$ (red diamonds), $s=1000$ (blue circles), and $s=100$ (green squares). The black solid line refers to the analytical semiclassical result (37) valid at large $s$ and $E/I_0\gtrsim 1/\sqrt{s}$. (Inset) The best simultaneous fit of the Lifshitz tail of three DOS by $\nu (E)\approx A\exp [-c_0{s}^{-1/4}\sqrt{I_0/E}\ln (I_0/E)]$ with the same $c_0=0.6375$ and $A=0.3425$.